PreprintCC BY 4.0arXiv
A counterexample to Scottish Book Problem 155
Abstract
We construct real Banach spaces X and Y and a bijection U: X→ Y which preserves all pairwise distances on every closed ball of radius 1/4 but is not a global isometry. The proof starts from an explicit injective map that preserves every distance at most 1/2 and shortens one longer distance. We then enlarge the source and target through a transfinite recursion that inserts every missing target point while preserving both injectivity and this distance threshold.
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BibTeX
@misc{ishiki2026counterexample,
title = {{A counterexample to Scottish Book Problem 155}},
author = {Yoshito Ishiki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29665},
eprint = {2609.29665},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29665},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A counterexample to Scottish Book Problem 155
AU - Yoshito Ishiki
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29665
UR - https://arxiv.org/abs/2609.29665
AB - We construct real Banach spaces X and Y and a bijection U: X→ Y which preserves all pairwise distances on every closed ball of radius 1/4 but is not a global isometry. The proof starts from an explicit injective map that preserves every distance at most 1/2 and shortens one longer distance. We then enlarge the source and target through a transfinite recursion that inserts every missing target point while preserving both injectivity and this distance threshold.
ER -